Subgroups — #12064883

Order 1280.

914 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#65214	C₂	1	2	yes	maximal
3	M₁₆×#168	C₂	1	8	yes	maximal, complemented
4	#10600517	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄×C₅×M₁₆	C₄	1	4	yes	—
8	C₄×C₅×M₁₆	C₂²	1	4	yes	—
9	C₂×C₅×#97	C₄	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₂²⋊#168	C₄	1	4	yes	—
12	#65214	C₄	1	4	yes	—
13	#65214	C₄	1	4	yes	—
14	#65214	C₂²	1	4	yes	—
15	#65214	C₂²	1	4	yes	—
16	C₂×C₄⋊#168	C₄	1	4	yes	—
17	C₂×C₄×#168	C₂²	1	4	yes	—
18	#64675	C₂²	1	8	yes	complemented
19	C₈×#168	C₂²	1	8	yes	complemented
20	#17314	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×M₁₆	C₈	1	16	yes	complemented
24	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	2	yes	—
33	C₂²×#168	C₂×C₄	1	8	yes	—
34	C₅×#97	C₂×C₄	1	8	yes	—
35	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
36	C₄×C₅×C₈	C₂³	1	4	yes	—
37	C₂²⋊#168	C₂×C₄	1	8	yes	—
38	C₂²⋊#168	C₂×C₄	1	8	yes	—
39	C₄⋊#168	C₂×C₄	1	8	yes	—
40	C₄⋊#168	C₂×C₄	1	8	yes	—
41	C₄×#168	C₂×C₄	1	8	yes	—
42	C₄×#168	C₂³	1	8	yes	—
43	C₂×#431	—	5	2	no	—
44	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
45	#1450	—	5	4	no	—
46	C₈×M₁₆	—	5	8	no	—
47	C₂⁴×C₅	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	4	4	no	—
49	C₂²×C₄×C₅	C₄²	1	2	yes	—
50	C₂²×C₄×C₅	C₄²	1	2	yes	—
51	C₂²×C₄×C₅	C₄²	1	2	yes	—
52	C₂²×C₄×C₅	C₄²	1	2	yes	—
53	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
60	C₅×M₁₆	C₂×C₈	1	32	yes	complemented
61	C₂×C₅×C₈	C₂×C₈	1	16	yes	complemented
62	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
63	C₄²×C₅	C₄²	1	1	yes	characteristic
64	C₄²×C₅	C₄²	1	2	yes	—
65	C₄²×C₅	C₄²	1	1	yes	characteristic
66	C₄²×C₅	C₄²	1	2	yes	—
67	C₄²×C₅	C₂²×C₄	1	2	yes	—
68	C₄²×C₅	C₂²×C₄	1	2	yes	—
69	C₄²×C₅	C₂²×C₄	1	2	yes	—
70	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₅	C₂²×C₄	1	2	yes	—
72	C₄²×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×#168	—	4	8	no	—
74	C₂×#168	C₂²×C₄	1	16	yes	—
75	C₂²×M₁₆	Dic₅	1	2	yes	—
76	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
77	#431	—	5	4	no	—
78	#431	—	5	4	no	—
79	#431	—	5	4	no	—
80	#431	—	5	4	no	—
81	C₂×C₄⋊C₈	—	5	4	no	—
82	C₂×C₂²⋊C₈	—	5	4	no	—
83	C₄×M₁₆	Dic₅	1	4	yes	—
84	C₄×M₁₆	C₂×D₅	1	4	yes	—
85	C₂×#97	Dic₅	1	2	yes	—
86	C₂×C₄×C₈	—	5	4	no	—
87	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
88	#322	—	5	8	no	—
89	C₈²	—	5	8	no	—
90	C₂³×C₅	—	4	2	no	—
91	C₂³×C₅	C₄×C₈	1	4	yes	—
92	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
93	C₂³×C₅	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	—	4	2	no	—
100	C₂×C₄×C₅	C₄×C₈	1	8	yes	—
101	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
102	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
109	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
111	C₅×C₈	C₂²×C₈	1	16	yes	—
112	#168	—	2	16	no	—
113	C₂³×C₄	C₄×D₅	1	2	yes	—
114	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
115	C₂×M₁₆	#168	1	16	yes	complemented
116	C₂×M₁₆	C₂×Dic₅	1	8	yes	—
117	C₂²×C₈	—	5	8	no	—
118	C₂²×C₈	C₂×Dic₅	1	4	yes	—
119	C₂×C₄²	C₄×D₅	1	2	yes	—
120	C₂×C₄²	C₄×D₅	1	2	yes	—
121	C₂×C₄²	C₄×D₅	1	2	yes	—
122	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
123	C₂×C₄²	C₂×Dic₅	1	2	yes	—
124	C₂×C₄²	C₂²×D₅	1	2	yes	—
125	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
126	C₄⋊C₈	—	5	8	no	—
127	C₄⋊C₈	—	5	8	no	—
128	C₂²⋊C₈	—	5	8	no	—
129	C₂²⋊C₈	—	5	8	no	—
130	#97	C₂×Dic₅	1	8	yes	—
131	C₄×C₈	—	5	8	no	—
132	C₄×C₈	—	5	8	no	—
133	C₄×C₈	C₂×Dic₅	1	4	yes	—
134	C₄×C₈	C₂²×D₅	1	4	yes	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	—	4	4	no	—
139	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
140	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
141	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
146	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
147	C₄×C₅	C₂³×C₈	1	2	yes	—
148	C₄×C₅	#568	1	4	yes	—
149	C₂⁴	C₄×Dic₅	1	1	yes	characteristic
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	C₂×#168	1	8	yes	—
152	C₂²×C₄	C₄×Dic₅	1	2	yes	—
153	C₂²×C₄	C₄×Dic₅	1	2	yes	—
154	C₂²×C₄	C₄×Dic₅	1	2	yes	—
155	C₂²×C₄	C₄×Dic₅	1	2	yes	—
156	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
158	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
159	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
160	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
162	M₁₆	C₂×#168	1	32	yes	complemented
163	C₂×C₈	—	5	16	no	—
164	C₂×C₈	—	20	8	no	—
165	C₂×C₈	C₂×#168	1	16	yes	complemented
166	C₂×C₈	C₂²×Dic₅	1	8	yes	—
167	C₄²	C₄×Dic₅	1	1	yes	characteristic
168	C₄²	C₄×Dic₅	1	2	yes	—
169	C₄²	C₄×Dic₅	1	1	yes	characteristic
170	C₄²	C₄×Dic₅	1	2	yes	—
171	C₄²	C₂×C₄×D₅	1	2	yes	—
172	C₄²	C₂×C₄×D₅	1	2	yes	—
173	C₄²	C₂×C₄×D₅	1	2	yes	—
174	C₄²	C₂²×Dic₅	1	1	yes	characteristic
175	C₄²	C₂²×Dic₅	1	2	yes	—
176	C₄²	C₂³×D₅	1	1	yes	characteristic
177	C₂×C₅	—	2	4	no	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
180	C₂×C₅	#2873	1	2	yes	—
181	C₂³	—	4	2	no	—
182	C₂³	C₄×#168	1	4	yes	—
183	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
184	C₂³	C₂×C₄×Dic₅	1	2	yes	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₄×#168	1	8	yes	—
192	C₂×C₄	C₄×#168	1	4	yes	—
193	C₂×C₄	C₂²×#168	1	4	yes	—
194	C₂×C₄	C₂²×#168	1	8	yes	—
195	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
196	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
197	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
198	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
199	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
201	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
202	C₈	—	10	16	no	—
203	C₈	C₂²×#168	1	16	yes	—
204	C₅	#17314	1	1	yes	characteristic, complemented, Sylow
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₂×C₄×#168	1	2	yes	—
210	C₂²	C₂×C₄×#168	1	4	yes	—
211	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	C₂×C₄×#168	1	2	yes	—
216	C₄	C₂×C₄×#168	1	4	yes	—
217	C₄	#66079	1	4	yes	—
218	C₄	C₂³×#168	1	2	yes	—
219	C₂	—	2	4	no	—
220	C₂	—	2	2	no	—
221	C₂	C₂²×C₄×#168	1	1	yes	characteristic
222	C₂	#10615478	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂²×C₄²
prime radical	C₂⁴×C₅	C₄²
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#17314
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. M₁₆×#168	C₂	semidirect
4. #10600517	C₂	semidirect
18. #64675	C₂²	semidirect
19. C₈×#168	C₂²	semidirect
23. C₂×C₅×M₁₆	C₈	semidirect
60. C₅×M₁₆	C₂×C₈	semidirect
61. C₂×C₅×C₈	C₂×C₈	semidirect
115. C₂×M₁₆	#168	semidirect
162. M₁₆	C₂×#168	semidirect
165. C₂×C₈	C₂×#168	semidirect
204. C₅	#17314	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₅×M₁₆	2
C₂×#65214	2
M₁₆×#168	2
#10600517	2
#17314	5